By Ginsparg P.
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As a result, this factor vanishes. It follows that the matrix element vanishes if two or more indices in the N-tuples b and c are interchanged. , bk 6 c and ci # b, or in detaiI In this c a e we c a n only obtain a non-zero value when the integral with h(zi)contains the two different orbitals #, and &. Thus, the sum over i only gives one contribution for the case i = 1 , whereas the sum over all permutations gives a non-zero contribution only for one particular permutation Po. This permutation is given by 'Inthis case we obtain We must now show that the second-quantization representation of to the same matrix elements: Po leads FUNDAMENTALS AND EXAMPLES If there are some i = b, and j = c, , we have where I, and tSindicate that the states b, and c, have been annihilated.
That the first variation of this expression with respect to the trial functions vanishes, (We should at this point recaIl that it is possible t o vary (Rev,) and (Imp,) independently. This possibility can however be rewritten as formally independent variations of rp, and&. ) We calcuIate the expectation vaIue of k in a Slater determinant by introducing second quantization ( cf. $. m r - m< * o 9 3 g 5 5-5 TEE HARTREE-POCK APPROXIMATION 54 In addition to the external potential u ( x ) and the classical potefitiaI energy dy p ( y ) u ( x , y), the Hartree-Fock potential contains a non-local part, the so-called exchange operator.
If we xi=1 xi€;. then abbreviate the interaction term in the Hartree-Fock potential as the single-particle operator j, we can then write the exact Hamiltohian in the form If we then treat theory 8' as a perturbation, weobtain in zeroth order perturbation N FUNDAMENTALS AND EXAMPLES and in first order From these expressions we see that Thus, the energy EHFwhich is optimized by using the variational principle agrees with the energy obtained from first order perturbation theory, whereas the sum over the single-particle energies E i only agrees with zeroth order perturbation theory.
Applied conformal field theory by Ginsparg P.